Science (English Medium) Class 12 - CBSE Question Bank Solutions

Evaluate the following as limit of sum:

`int_0^2 "e"^x "d"x`

Evaluate the following:

`int_0^2 ("d"x)/("e"^x + "e"^-x)`

Evaluate the following:

`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`

Evaluate the following:

`int_0^1 (x"d"x)/sqrt(1 + x^2)`

Evaluate the following:

`int_0^pi x sin x cos^2x "d"x`

Evaluate the following:

`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))`  (Hint: Let x = sin θ)

Evaluate the following:

`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2)  "d"x`

The value of `int_(-pi)^pi sin^3x cos^2x  "d"x` is ______.

Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`

Prove that in a ∆ABC,  `sin"A"/"a" = sin"B"/"b" = sin"C"/"c"`, where a, b, c represent the magnitudes of the sides opposite to vertices A, B, C, respectively.

The magnitude of the vector `6hat"i" + 2hat"j" + 3hat"k"` is ______.

A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.

Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.

Prove that in any triangle ABC, cos A = `("b"^2 + "c"^2 - "a"^2)/(2"bc")`, where a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.

The vector in the direction of the vector `hat"i" - 2hat"j" + 2hat"k"` that has magnitude 9 is ______.

The magnitude of the vector `6hati - 2hatj + 3hatk` is ______.

Read the following passage and answer the questions given below:

1. What is the magnitude of the force of Team A ?
2. Which team will win the game?
3. Find the magnitude of the resultant force exerted by the teams.
   OR
   In what direction is the ring getting pulled?

Evaluate : `int_0^3dx/(9+x^2)`

If a unit vector `veca` makes angles `pi/3` with `hati,pi/4` with `hatj` and acute angles θ with ` hatk,` then find the value of θ.

Write the value of `vec a .(vecb xxveca)`